What kind of check endorsement is signed by the person it is made out to but has instructions to pay the check to someone else?

This type of endorsement is known as a "third-party endorsement." In this scenario, the payee signs the back of the check and adds instructions to pay the funds to another person, effectively transferring the check's payment to that individual. It is essential for the endorsement to clearly state the new payee's name to ensure proper processing by the bank.